Conical Free Disposal Hull estimators of directional distances and Luenberger productivity indices for general technologies

Abstract

Directional distances are a popular tool in productivity and efficiency analysis due to their versatility in evaluating the distance of Decision Making Units (DMU) to the efficient frontier of the production set. The theoretical and statistical properties of these measures are well-established in various contexts. However, the measurement of directional distances to the cone spanned by the attainable set has not yet been explored. This cone is necessary to define the Luenberger indices for general technologies. This paper aims to fill this gap by presenting a method for defining and estimating directional distances to this cone, applicable to general technologies without imposing convexity. We also discuss the statistical properties of these measures, enabling us to measure distances to non-convex attainable sets under Constant Returns to Scale (CRS), as well as measure and estimate Luenberger productivity indices and their decompositions for general technologies. In addition, we provide a detailed description of how to make inferences on these indices. Finally, we offer simulated data and a practical example of inference on Luenberger productivity indices and their decompositions using a well-known real data set